Example 3.13 New examples of varieties with the weak-Hilbert property can be given by taking products. For example, let \(X\)  be the K3 surface defined by \(x^4+y^4=z^4+w^4\)  in \(\mathbb {P}_{3,\mathbb {Q}}\) , and note that \(X\)  has the weak-Hilbert property over \(\mathbb {Q}\)  (see [1]}). Let \(A\)  be an abelian variety over \(\mathbb {Q}\)  with \(A(\mathbb {Q})\)  dense, so that \(A\)  has the weak-Hilbert property (Theorem REF ). Then, for every positive integer \(m\) , since \(\mathbb {P}_{n,\mathbb {Q}}\)  has the (weak-)Hilbert property over \(\mathbb {Q}\) , the variety \(A\times \mathbb {P}_{n,\mathbb {Q}}\times X^{m}\)  has the weak-Hilbert property over \(\mathbb {Q}\)  by Theorem REF .

